Best approach to rigging an alligator
Rigify is able to do the job. Rigify is a modular rig system to create bones and controls with automatic or manual FK/IK. Keep in mind the settings for Rigify differ weather you're on edit mode (for the Armature), the pose mode (for bone constraints) and animation (drivers). There is a complete guide showing you step by step how to learn Rigify. The only guide I seriously recommend: https://gumroad.com/a/189101171
On Lesson 17 there is a full rig exercise with a dragon head, that way you can also learn how to rig blinking eyes for reptiles.
If you need further -specific setup answers- please post them here.